(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the parametric equations of the line tangent to the curve of intersection of the paraboloid

z = x² + y² and the ellipsoid 4x² + y² + z² = 9 at the point ( -1, 1, 2 ).

2. Relevant equations

Probable use of the gradient vector (as this is the chapter we are in)

3. The attempt at a solution

I have found what I believe to be the curve of intersection by setting the equations of the surfaces equal to each other:

x² + y² - z = 4x² + y² + z² - 9

9 = 3x² + z² + z

F(x,y,z) = 3x² + z² + z - 9 (as a function)

And the corresponding gradient vector:

< 6x, 0, 2z + 1 >

But I cannot figure out how to convert this from cartesian to polar (or if I even need to) and get the correct answer. I've tried several paths (i.e. substituting x=rcosσ) but my answer never matches the expected answer, which contains no trigonometry. Does the correct approach involve substituting x² + y² for z? This seems to be the most logical approach but I haven't been able to apply it correctly.

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# Homework Help: Tangent line of curve of intersection

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